Two's complement

Find -50

Compute +50
-128 64  32  16  8  4  2  1
 0    0   1   1  0  0  1  0 = 50
=============================
 1    1   0   0  1  1  0  1  reverse bits (1's complement)
			 +1  add one
----------------------------
 1    1   0   0  1  1  1  0 = -50 in two's complement


Base 2 addition
 0    1   0   1
+0   +0  +1  +1
---  --- --- ---
 0    1   1  10

 1  Example: 
 1           0  1  1  0  1  1  1  0
+1           0  0  1  1  0  1  1  0
---          ----------------------
11           1  0  1  0  0  1  0  0


Floating Point

-5.33333 * 10^-6 = -.0000053333

The Mantissa of a Normalized Floating Point Number -1 < N < 1

-5.3333 * 10^-6 = -.53333 * 10^-5

In Base 2 we only have 0s and 1s.  Therefore numbers look like

-.00101011 * 2^-3 = -.00000101011

A normalized floating point base 2 number's mantissa begins with 1.

-.00000101011 = -.101011 * 2^-5

Since the first bit is ALWAYS 1, it need not be stored.
What we store is

Mantissa = 010110, sign = -1 and exponent -5
The first 1 bit is the "hidden bit".

Next we must know how many bits are allocated to
1. The mantissa
2. exponent

Let's consider the number 2,000,000,000 which is approx. 2^31
stored as a floating point number.  This is 1 * 2^31. 
Therefore the exponent of 2 must be 31.
31 = 11111 = 16+8+4++1

16 bit floating point representation COULD be (we'll make this up)
Let's use 6 bits for the exponent.
We must use one bit for the sign.
That leaves 9 bits for the mantissa.